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Porcupine is a linear temperament in the porcupine family that tempers out 250/243, the porcupine comma, and whose generator is somewhere around 160-165 cents. It can be thought of as a 5-limit, 7-limit, or 11-limit temperament, or a 2.3.5.11 subgroup temperament. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.

The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)^2 equivalent to (6/5)^3. In perhaps more familiar musical terms, this means two "perfect fourths" equals three "minor thirds". As a consequence of this, 4/3 is divided into 3 equal parts, and 6/5 is divided into 2 of those same equal parts. This is obviously in stark contrast to 12edo, and to meantone, in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales.

Porcupine symmetric minor scale, containing two equal tetrachords with a major wholetone between them. (Tuning in 22edo)

Interval chain

Main article: Porcupine intervals

Generators	Cents	Ratios	Ups and Downs notation	Generators	2/1 inverse	Ratios	Ups and Downs notation
0	0.00	1/1	P1	0	1200.00	2/1	P8
1	162.75	12/11~11/10~10/9	vM2 = ^^m2	-1	1037.25	9/5~20/11~11/6	^m7 = vvM7
2	325.50	6/5~11/9	^m3 = vvM3	-2	874.50	18/11~5/3	vM6 = ^^m6
3	488.25	4/3	P4	-3	711.75	3/2	P5
4	651.00	16/11~22/15	v5 = ^^d5	-4	549.00	15/11~11/8	^4 = vvA4
5	813.75	8/5	^m6 = vvM6	-5	386.25	5/4	vM3 = ^^m3
6	976.50	7/4~16/9	m7	-6	223.50	9/8~8/7	M2
7	1139.25	48/25~160/81	v8 = ^^d8	-7	60.75	81/80~25/24	^1 = vvA1
8	102.00	16/15~21/20	^m2 = vvM2	-8	1098.00	40/21~15/8	vM7 = ^^m7
9	264.75	7/6	m3	-9	935.25	12/7	M6
10	427.50	14/11	v4 = ^^d4	-10	772.50	11/7	^5 = vvA5
11	590.25	7/5	^d5 = vv5	-11	609.75	10/7	vA4 = ^^4
12	753.00	14/9	m6	-12	447.00	9/7	M3

The specific tuning shown is the full 11-limit POTE tuning, but of course there is a range of acceptible porcupine tunings that includes generators as small as 160 cents (15edo) and as large as 165.5 cents (29edo). (However, the 29edo patent val does not support 11-limit porcupine proper, not annihilating 64/63.)
12/11, 11/10, and 10/9 are all represented by the same interval, the generator. This makes chords such as 8:9:10:11:12 exceptionally common and easy to find.


8:9:10:11:12 chord, in just intonation. All intervals are slightly different.	Porcupine-tempered 8:9:10:11:12 chord, in 22edo. Except the first, the intervals are the same.	Porcupine-tempered 8:9:10:11:12 chord, in 29edo. Except the first, the intervals are the same.

The 11/9 interval, usually considered a "neutral third", is in porcupine identical to the 6/5 "minor third". This means that the 27/20 "acute fourth" of the JI diatonic scale is equivalent to 11/8 (rather than becoming 4/3 as in meantone).
The characteristic small interval of porcupine, which is 60.75 cents in this tuning but can range from <50 to 80 cents in general, represents both 25/24 and 81/80.

Spectrum of Porcupine Tunings by Eigenmonzos

Eigenmonzo	Neutral Second
13/12	138.573
13/11	144.605
12/11	150.637
13/10	151.405
6/5	157.821
15/13	158.710
18/13	159.154
2\15	160.000
8/7	161.471
14/11	161.751
7/5	162.047
5\37	162.162
11/8	162.171 13- and 15-limit minimax
8\59	162.712
5/4	162.737 5-limit minimax
15/14	162.897
7/6	162.986
3\22	163.636
9/7	163.743 7- 9- and 11-limit minimax
16/15	163.966
7\51	164.706
11/10	165.004
4\29	165.517
15/11	165.762
4/3	166.015
14/13	166.037
11/9	173.704
16/13	179.736
10/9	182.404

[8/5 12/7] eigenmonzos: porcupinewoo15 porcupinewoo22

Spectrum of Porcupinefish Tunings

12/11	150.637
6/5	157.821
2\15	160.000
18/13	160.307
15/13	160.860
8/7	161.471
13/12	161.531
14/11	161.751
7/5	162.047
14/13	162.100
13/10	162.149
5\37	162.162
11/8	162.171
16/13	162.322
13/11	162.368 13- and 15-limit minimax
8\59	162.712
5/4	162.737
15/14	162.897
7/6	162.986
3\22	163.636
9/7	163.743
16/15	163.966
7\51	164.706
11/10	165.004
4\29	165.517
15/11	165.762
4/3	166.015
11/9	173.704
10/9	182.404

History

Porcupine temperament/scales were discovered by Dave Keenan, but didn't have a name until Herman Miller mentioned that his Mizarian Porcupine Overture in 15-tET had a section that pumps the 250:243 comma. Although this music did not use a Porcupine MOS or MODMOS (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of DE scales was fully evident. It was clear that even though Herman's piece was in 15, 22 was a porcupine tuning par excellence, and that was an interesting development in itself.

Musical examples

"Mizarian Porcupine Overture", Herman Miller, 1999. (15edo, namesake of the temperament)
"Glassic", Paul Erlich, 22edo (at least the beginning part is in porcupine).
"Night on Porcupine Mountain", Gene Ward Smith and Modest Mussorgsky, 22edo.
"being a", Andrew Heathwaite, 2010, 22edo, mode 3 1 3 3 3 3 3 3 of Porcupine[8].
Playing Gently with Miller's Porcupine, Chris Vaisvil
15 Porcupines in India, Sarangi, Tambura and Sitar improvisation by Chris Vaisvil
15 Quills piano solo by Chris Vaisvil
Prickly Side of Love - rock band in Porcupine Temperament with vocals by Chris Vaisvil
Porcupine Organ Composition by Chris Vaisvil
Among Other Things 2 by Petr Pařízek
Porcupine Comma Pump, by Jake Freivald
Life on Mars by Omega9